logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premis ...

, a normal modal logic is a set ''L'' of modal formulas such that ''L'' contains: * All propositional tautologies; * All instances of the Kripke schema:

\Box(A\to B)\to(\Box A\to\Box B)

and it is closed under: * Detachment rule ('' modus ponens''):

A\to B, A \in L

implies

B \in L

; * Necessitation rule:

A \in L

implies

\Box A \in L

. The smallest logic satisfying the above conditions is called K. Most modal logics commonly used nowadays (in terms of having philosophical motivations), e.g.

C. I. Lewis Clarence Irving Lewis (April 12, 1883 – February 3, 1964), usually cited as C. I. Lewis, was an American academic philosopher. He is considered the progenitor of modern modal logic and the founder of conceptual pragmatism. First a noted log ...

's S4 and S5, are normal (and hence are extensions of K). However a number of

deontic In moral philosophy, deontological ethics or deontology (from Greek: + ) is the normative ethical theory that the morality of an action should be based on whether that action itself is right or wrong under a series of rules and principles, ra ...

and

epistemic logic Epistemic modal logic is a subfield of modal logic that is concerned with reasoning about knowledge. While epistemology has a long philosophical tradition dating back to Ancient Greece, epistemic logic is a much more recent development with applic ...

s, for example, are non-normal, often because they give up the Kripke schema. Every normal modal logic is

regular The term regular can mean normal or in accordance with rules. It may refer to: People * Moses Regular (born 1971), America football player Arts, entertainment, and media Music * "Regular" (Badfinger song) * Regular tunings of stringed instrum ...

and hence

classical Classical may refer to: European antiquity *Classical antiquity, a period of history from roughly the 7th or 8th century B.C.E. to the 5th century C.E. centered on the Mediterranean Sea *Classical architecture, architecture derived from Greek and ...

Common normal modal logics

The following table lists several common normal modal systems. The notation refers to the table at Kripke semantics § Common modal axiom schemata. Frame conditions for some of the systems were simplified: the logics are ''sound and complete'' with respect to the frame classes given in the table, but they may ''correspond'' to a larger class of frames.

References

*Alexander Chagrov and Michael Zakharyaschev, ''Modal Logic'', vol. 35 of Oxford Logic Guides, Oxford University Press, 1997. {{Logic-stub Modal logic